Solution of Triangles: Cyclic Quadrilaterals

Cyclic quadrilaterals are a special type of quadrilateral that can be inscribed in a circle, meaning that all four vertices of the quadrilateral lie on the circumference of the circle. The study of cyclic quadrilaterals is an important part of trigonometry and geometry, especially when it comes to solving triangles, as it involves understanding the relationships between the sides and angles of the quadrilateral.

Properties of Cyclic Quadrilaterals

Before we dive into the solution of triangles within cyclic quadrilaterals, let's review some key properties that define them:

• Opposite Angles: The sum of the opposite angles of a cyclic quadrilateral is 180 degrees (supplementary).
• Exterior Angle: The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.
• Ptolemy's Theorem: For a cyclic quadrilateral with sides a, b, c, and d, and diagonals p and q, the product of the diagonals is equal to the sum of the products of opposite sides: ( p \cdot q = a \cdot c + b \cdot d ).
• Brahmagupta's Formula: If a cyclic quadrilateral is inscribed in a circle of radius R and has sides a, b, c, and d, the area (A) of the quadrilateral can be found using the formula: ( A = \sqrt{(s-a)(s-b)(s-c)(s-d)} ), where ( s ) is the semiperimeter ( s = \frac{a+b+c+d}{2} ).

Differences and Important Points

Property	Cyclic Quadrilateral	General Quadrilateral
Angles	Opposite angles are supplementary.	No such restriction.
Diagonals	Ptolemy's theorem applies.	Ptolemy's theorem does not apply.
Circumcircle	All vertices lie on a single circle.	Not all vertices may lie on a single circle.
Area	Brahmagupta's formula can be used.	Different formulas are used, depending on the type of quadrilateral.

Formulas

Here are some formulas that are particularly useful when working with cyclic quadrilaterals:

• Opposite Angles: ( \angle A + \angle C = 180^\circ ) and ( \angle B + \angle D = 180^\circ )
• Ptolemy's Theorem: ( p \cdot q = a \cdot c + b \cdot d )
• Brahmagupta's Formula: ( A = \sqrt{(s-a)(s-b)(s-c)(s-d)} )

Examples

Example 1: Opposite Angles

Given a cyclic quadrilateral ABCD, where ( \angle A = 70^\circ ) and ( \angle B = 85^\circ ), find the measures of ( \angle C ) and ( \angle D ).

Solution:

Since opposite angles are supplementary:

( \angle C = 180^\circ - \angle A = 180^\circ - 70^\circ = 110^\circ ) ( \angle D = 180^\circ - \angle B = 180^\circ - 85^\circ = 95^\circ )

Example 2: Ptolemy's Theorem

Given a cyclic quadrilateral ABCD with sides a = 5, b = 7, c = 6, and d = 8, and diagonal p = 9, find the length of the other diagonal q.

Solution:

Using Ptolemy's theorem:

( p \cdot q = a \cdot c + b \cdot d ) ( 9 \cdot q = 5 \cdot 6 + 7 \cdot 8 ) ( 9 \cdot q = 30 + 56 ) ( 9 \cdot q = 86 ) ( q = \frac{86}{9} )

Example 3: Brahmagupta's Formula

Given a cyclic quadrilateral ABCD with sides a = 4, b = 5, c = 6, and d = 7, find the area of the quadrilateral.

Solution:

First, calculate the semiperimeter:

( s = \frac{a+b+c+d}{2} = \frac{4+5+6+7}{2} = \frac{22}{2} = 11 )

Now, use Brahmagupta's formula to find the area:

( A = \sqrt{(s-a)(s-b)(s-c)(s-d)} ) ( A = \sqrt{(11-4)(11-5)(11-6)(11-7)} ) ( A = \sqrt{7 \cdot 6 \cdot 5 \cdot 4} ) ( A = \sqrt{840} )

The area of the cyclic quadrilateral is ( \sqrt{840} ) square units.

By understanding the properties, formulas, and examples provided, students can effectively solve problems related to cyclic quadrilaterals in their exams.